So consider the following laws of motion in an inertial system : $$ m_k \partial_{tt} x_k = - \partial_{x_k} V(x_1, .., x_k)$$ where $x_i \in \mathbb{R}^3$ is a particle of mass $m_i$. By principle of relativity, we find that we must have $G$-invariance of $V$, where $G$ is the galilean group of transformations $x \to Rx +b$ acting on $\mathbb{R}^{3n}$ diagonally, ie. $$V(Rx_1+b, .., Rx_k+b) = V(x_1,...,x_k)$$ for any $R \in O(3), b \in \mathbb{R}^3$.
From this we can deduce, in the case $k=2$, that $V = V(|x_2-x_1|)$. How can we make further conclusions for $k$ particles ? How can we even know the number of degrees of freedom left in the argument of V ? In other words, what do we know about the structure of $\mathbb{R}^{3n} / G$ ? As a physics student coming from maths, is there a way for me to identify a change of variable $(x_1, .., x_k) \to (\alpha_1, .., \alpha_r)$ that would live in $\mathbb{R}^{3n} / G$ and be relevant physically ?
I tried to get insight from Burnside's lemma, but couldn't really, because the formula works for finite groups and sets and does not extend (obviously) to insight on dimensionality of the quotient set.